Properties

Label 1-2001-2001.77-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.997 + 0.0672i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.670 + 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (0.806 + 0.591i)8-s + (0.242 + 0.970i)10-s + (0.998 − 0.0611i)11-s + (−0.794 − 0.607i)13-s + (−0.830 + 0.557i)14-s + (−0.979 + 0.202i)16-s + (0.540 + 0.841i)17-s + (0.994 − 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (0.983 − 0.182i)26-s + ⋯
L(s)  = 1  + (−0.670 + 0.742i)2-s + (−0.101 − 0.994i)4-s + (0.557 − 0.830i)5-s + (0.970 + 0.242i)7-s + (0.806 + 0.591i)8-s + (0.242 + 0.970i)10-s + (0.998 − 0.0611i)11-s + (−0.794 − 0.607i)13-s + (−0.830 + 0.557i)14-s + (−0.979 + 0.202i)16-s + (0.540 + 0.841i)17-s + (0.994 − 0.101i)19-s + (−0.882 − 0.470i)20-s + (−0.623 + 0.781i)22-s + (−0.377 − 0.925i)25-s + (0.983 − 0.182i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.997 + 0.0672i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.997 + 0.0672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.640163185 + 0.05518616995i\)
\(L(\frac12)\) \(\approx\) \(1.640163185 + 0.05518616995i\)
\(L(1)\) \(\approx\) \(1.045755559 + 0.1225813569i\)
\(L(1)\) \(\approx\) \(1.045755559 + 0.1225813569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.670 + 0.742i)T \)
5 \( 1 + (0.557 - 0.830i)T \)
7 \( 1 + (0.970 + 0.242i)T \)
11 \( 1 + (0.998 - 0.0611i)T \)
13 \( 1 + (-0.794 - 0.607i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.994 - 0.101i)T \)
31 \( 1 + (0.965 - 0.262i)T \)
37 \( 1 + (-0.359 - 0.933i)T \)
41 \( 1 + (0.989 + 0.142i)T \)
43 \( 1 + (-0.965 - 0.262i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.0203 + 0.999i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.0407 + 0.999i)T \)
67 \( 1 + (0.0611 - 0.998i)T \)
71 \( 1 + (0.488 - 0.872i)T \)
73 \( 1 + (0.122 - 0.992i)T \)
79 \( 1 + (-0.202 + 0.979i)T \)
83 \( 1 + (-0.685 + 0.728i)T \)
89 \( 1 + (0.396 + 0.917i)T \)
97 \( 1 + (-0.505 - 0.862i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.01792880257443865140681544365, −19.00328090284567291447193803902, −18.63834249037624001703601346731, −17.7078723000809665089032160192, −17.33973785180889921718880889506, −16.65718507613739691831718115899, −15.66373710636955174289219701787, −14.446251284799218463780111172731, −14.18892483224288020373481736695, −13.430482165578130734342057259341, −12.17942920168866824501140158347, −11.638998726531739763485567188709, −11.18316242576921016407933058735, −10.074912917700286499855984854, −9.75217174195664514258424062368, −8.912465516633703790751721535822, −7.93306639479746386878108692350, −7.194152962195480298049942679545, −6.644862433060171060446085450112, −5.295453329264873250284934151571, −4.46518381845489244655391447, −3.48178505782538198564893949509, −2.65994460973051437549819139319, −1.79504322499167191249749974903, −1.04426073275565002539619967266, 0.917773824827270935549958718483, 1.50164883861093452703420718428, 2.49907106583939811733206419619, 4.09178364434060443630223846978, 4.91047811142209089857849802422, 5.58358529457710036123011791586, 6.18054810781446896007006479739, 7.36173321725175025593855473021, 7.95181044572485682047761180180, 8.73538979964682578159095310905, 9.33976440778388486212941881953, 10.058354060703720268164520644304, 10.87138326489463794516341804956, 11.902412673606034073880368160174, 12.46816517699899803600198882228, 13.72919913311414897327345675888, 14.14479815603925883232868196670, 14.97887626978429613392701249837, 15.54566557684885075237573776707, 16.70909342109229067410471819914, 16.9441205734156675996851294483, 17.74226452614680251127446061972, 18.128406709514805708221792208350, 19.28168953038710521730597464404, 19.812963674423999005601646995965

Graph of the $Z$-function along the critical line